The Modulation of Multiple Phases Leading to the Modified KdV Equation

This paper seeks to derive the modified KdV (mKdV) equation using a novel approach from systems generated from abstract Lagrangians that possess a two-parameter symmetry group. The method to do uses a modified modulation approach, which results in the mKdV emerging with coefficients related to the conservation laws possessed by the original Lagrangian system. Alongside this, an adaptation of the method of Kuramoto is developed, providing a simpler mechanism to determine the coefficients of the nonlinear term. The theory is illustrated using two examples of physical interest, one in stratified hydrodynamics and another using a coupled Nonlinear Schr\"odinger model, to illustrate how the criterion for the mKdV equation to emerge may be assessed and its coefficients generated.Comment: 35 pages, 5 figure

Mesa-type patterns in the one-dimensional Brusselator and their stability

The Brusselator is a generic reaction-diffusion model for a tri-molecular chemical reaction. We consider the case when the input and output reactions are slow. In this limit, we show the existence of $K$-periodic, spatially bi-stable structures, \emph{mesas}, and study their stability. Using singular perturbation techniques, we find a threshold for the stability of $K$ mesas. This threshold occurs in the regime where the exponentially small tails of the localized structures start to interact. By comparing our results with Turing analysis, we show that in the generic case, a Turing instability is followed by a slow coarsening process whereby logarithmically many mesas are annihilated before the system reaches a steady equilibrium state. We also study a ``breather''-type instability of a mesa, which occurs due to a Hopf bifurcation. Full numerical simulations are shown to confirm the analytical results.Comment: to appear, Physica

Existence and stability of hole solutions to complex Ginzburg-Landau equations

We consider the existence and stability of the hole, or dark soliton, solution to a Ginzburg-Landau perturbation of the defocusing nonlinear Schroedinger equation (NLS), and to the nearly real complex Ginzburg-Landau equation (CGL). By using dynamical systems techniques, it is shown that the dark soliton can persist as either a regular perturbation or a singular perturbation of that which exists for the NLS. When considering the stability of the soliton, a major difficulty which must be overcome is that eigenvalues may bifurcate out of the continuous spectrum, i.e., an edge bifurcation may occur. Since the continuous spectrum for the NLS covers the imaginary axis, and since for the CGL it touches the origin, such a bifurcation may lead to an unstable wave. An additional important consideration is that an edge bifurcation can happen even if there are no eigenvalues embedded in the continuous spectrum. Building on and refining ideas first presented in Kapitula and Sandstede (Physica D, 1998) and Kapitula (SIAM J. Math. Anal., 1999), we show that when the wave persists as a regular perturbation, at most three eigenvalues will bifurcate out of the continuous spectrum. Furthermore, we precisely track these bifurcating eigenvalues, and thus are able to give conditions for which the perturbed wave will be stable. For the NLS the results are an improvement and refinement of previous work, while the results for the CGL are new. The techniques presented are very general and are therefore applicable to a much larger class of problems than those considered here.Comment: 41 pages, 4 figures, submitte

Existence and Stability of a Spike in the Central Component for a Consumer Chain Model

We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in \cite{gm,schn1} for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect: if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction-diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir

Phase Slips and the Eckhaus Instability

We consider the Ginzburg-Landau equation, $\partial_t u= \partial_x^2 u + u - u|u|^2$, with complex amplitude $u(x,t)$. We first analyze the phenomenon of phase slips as a consequence of the {\it local} shape of $u$. We next prove a {\it global} theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped with these results, we proceed to prove the corresponding phenomena for the fourth order Swift-Hohenberg equation, of which the Ginzburg-Landau equation is the amplitude approximation. This sheds light on how one should deal with local and global aspects of phase slips for this and many other similar systems.Comment: 22 pages, Postscript, A

Orbital stability of periodic waves for the nonlinear Schroedinger equation

The nonlinear Schroedinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure

Global consequences of afforestation and bioenergy cultivation on ecosystem service indicators

Land management for carbon storage is discussed as being indispensable for climate change mitigation because of its large potential to remove carbon dioxide from the atmosphere, and to avoid further emissions from deforestation. However, the acceptance and feasibility of land-based mitigation projects depends on potential side effects on other important ecosystem functions and their services. Here, we use projections of future land use and land cover for different land-based mitigation options from two land-use models (IMAGE and MAgPIE) and evaluate their effects with a global dynamic vegetation model (LPJ-GUESS). In the land-use models, carbon removal was achieved either via growth of bioenergy crops combined with carbon capture and storage, via avoided deforestation and afforestation, or via a combination of both. We compare these scenarios to a reference scenario without land-based mitigation and analyse the LPJ-GUESS simulations with the aim of assessing synergies and trade-offs across a range of ecosystem service indicators: carbon storage, surface albedo, evapotranspiration, water runoff, crop production, nitrogen loss, and emissions of biogenic volatile organic compounds. In our mitigation simulations cumulative carbon storage by year 2099 ranged between 55 and 89 GtC. Other ecosystem service indicators were influenced heterogeneously both positively and negatively, with large variability across regions and land-use scenarios. Avoided deforestation and afforestation led to an increase in evapotranspiration and enhanced emissions of biogenic volatile organic compounds, and to a decrease in albedo, runoff, and nitrogen loss. Crop production could also decrease in the afforestation scenarios as a result of reduced crop area, especially for MAgPIE land-use patterns, if assumed increases in crop yields cannot be realized. Bioenergy-based climate change mitigation was projected to affect less area globally than in the forest expansion scenarios, and resulted in less pronounced changes in most ecosystem service indicators than forest-based mitigation, but included a possible decrease in nitrogen loss, crop production, and biogenic volatile organic compounds emissions

Citrullinated human and murine MOG_35–55 display distinct biophysical and biochemical behavior

The peptide spanning residues 35 to 55 of the protein myelin oligodendrocyte glycoprotein (MOG) has been studied extensively in its role as a key autoantigen in the neuroinflammatory autoimmune disease multiple sclerosis. Rodents and nonhuman primate species immunized with this peptide develop a neuroinflammatory condition called experimental autoimmune encephalomyelitis, often used as a model for multiple sclerosis. Over the last decade, the role of citrullination of this antigen in the disease onset and progression has come under increased scrutiny. We recently reported on the ability of these citrullinated MOG35–55 peptides to aggregate in an amyloid-like fashion, suggesting a new potential pathogenic mechanism underlying this disease. The immunodominant region of MOG is highly conserved between species, with the only difference between the murine and human protein, a polymorphism on position 42, which is serine in mice and proline for humans. Here, we show that the biophysical and biochemical behavior we previously observed for citrullinated murine MOG35–55 is fundamentally different for human and mouse MOG35–55. The citrullinated human peptides do not show amyloid-like behavior under the conditions where the murine peptides do. Moreover, we tested the ability of these peptides to stimulate lymphocytes derived from MOG immunized marmoset monkeys. While the citrullinated murine peptides did not produce a proliferative response, one of the citrullinated human peptides did. We postulate that this unexpected difference is caused by disparate antigen processing. Taken together, our results suggest that further study on the role of citrullination in MOG-induced experimental autoimmune encephalomyelitis is necessary.</p